WEBVTT 00:00:05.606 --> 00:00:10.864 Let's take a look at the sum and difference formulas. Here I've listed on the board the sum and difference 00:00:10.866 --> 00:00:16.369 formulas for cosine, sine, and tangent. Let's see how we can use these sum and difference formulas to 00:00:16.371 --> 00:00:24.369 find exact values. In this example, I want to find the exact value of the sine of 7 pi over 12, and I'm going 00:00:24.371 --> 00:00:32.385 to use a sum formula. So the problem is 7 pi over 12 of course is not one of your special angles. We want 00:00:32.387 --> 00:00:40.635 to rewrite 7 pi over 12 as the sum of two angles, or the difference of two angles, that are special. So I'll 00:00:40.637 --> 00:00:53.886 write this as the sine of 4 pi over 12 plus 3 pi over 12. Now of course that adds up to 7 pi over 12, and what's 00:00:53.888 --> 00:01:01.887 very important about that is that when you reduce these you get pi over 3, which of course is a special 00:01:01.889 --> 00:01:08.386 angle, and pi over 4, which of course is another special angle, so we can find the exact value. You couldn't 00:01:08.388 --> 00:01:16.634 rewrite this, say, as 5 pi over 12 plus 2 pi over 12 because 5 pi over 12 is not a special angle. So it's very 00:01:16.636 --> 00:01:23.887 important that when you rewrite it these two angles are special. So now we have the sine of an angle plus 00:01:23.889 --> 00:01:31.383 an angle. Well we can use this identity right here. So in this case, my alpha is pi over 3, and my beta is 00:01:31.385 --> 00:01:42.704 pi over 4. So using this identity, I can rewrite this as the sine of pi over 3 cosine pi over 4 plus the cosine 00:01:42.706 --> 00:01:53.199 of pi over 3 times the sine of pi over 4. Don't skip the step where you write the identity. Now we've got 00:01:53.201 --> 00:01:58.722 it down to trig functions of special angles, and you should know these values very well at this point. 00:01:58.7244 --> 00:02:09.719 Let's plug in those values. The sine of pi over 3, remember, is square root of 3 over 2. The cosine of pi over 4, 00:02:09.721 --> 00:02:21.242 recall, is 1 over the square root of 2. The cosine of pi over 3 is 1/2, and finally, the sine of pi over 4 is 00:02:21.244 --> 00:02:30.253 1 over the square root of 2. So let's see what we get. We get the square root of 3 over 2 square roots of 2 00:02:30.255 --> 00:02:36.758 plus 1 over 2 square roots of 2. Since we have a common denominator, we can add and get the square 00:02:36.760 --> 00:02:47.018 root of 3 plus 1 over 2 square roots of 2. So the sine of 7 pi over 12, the exact value, 00:02:47.020 --> 00:02:54.517 is square root of 3 plus 1 over 2 square roots of 2. Let's take a look at another example. 00:02:57.200 --> 00:03:03.514 In this example, we want to find the exact value of the tangent of 17 pi over 12 without the use of a 00:03:03.516 --> 00:03:13.294 calculator. This time we're going to think of two numbers that either add or subtract to get 17 pi over 12. 00:03:13.296 --> 00:03:24.292 Well, let's see. 17 pi over 12 could be written as 8 pi over 12 plus 9 pi over 12. That certainly adds 00:03:24.294 --> 00:03:32.293 up to 17, and if I reduce each of these, let's see. That'll be 2 pi over 3 and 3 pi over 4. Those are both 00:03:32.295 --> 00:03:41.293 special. But you could also write it, there are lots of other ways, you could write it as 21 pi over 12 minus 00:03:41.295 --> 00:03:50.540 4 pi over 12. If I reduce each of these, let's see what we get. Let's see, 3, that would be 7 pi over 4 00:03:50.542 --> 00:03:57.790 minus pi over 3. So the way you rewrite it is not unique. Oftentimes there's more than one way. Let's go 00:03:57.792 --> 00:04:09.852 ahead and use this way. So we're going to rewrite this as the tangent of 21 pi over 12 minus 4 pi over 12, 00:04:09.854 --> 00:04:21.361 and that's 7 pi over 4 minus pi over 3. Ok, so now we've got it written as the tangent of alpha minus beta, 00:04:21.363 --> 00:04:31.615 so we can use this identity right here. My alpha is 7 pi over 4. My beta is pi over 3. So that's going to give 00:04:31.617 --> 00:04:44.883 us tangent of 7 pi over 4 minus tangent of pi over 3 all over 1 plus tangent of 7 pi over 4 times the 00:04:44.885 --> 00:04:51.885 tangent of pi over 3. Ok, now we just need to find those values. Again, these should be values that you 00:04:51.887 --> 00:05:00.637 know very well at this point. The tangent of 7 pi over 4, well the reference angle is pi over 4. Tangent of 00:05:00.639 --> 00:05:06.886 pi over 4 is 1, but remember 7 pi over 4 is in quadrant four. The tangent is negative, so this will be 00:05:06.888 --> 00:05:21.137 negative 1. The tangent of pi over 3 is the square root of 3. Plugging in those values, we get negative 1 00:05:21.139 --> 00:05:37.630 minus the square root of 3 over 1 minus the square root of 3. So this is equal to the tangent of 17 pi over 12. 00:05:39.650 --> 00:05:46.688 In this example, we want to find the exact value of this expression without the use of a calculator. 00:05:46.690 --> 00:05:52.194 So we look at it. We notice these angles are not special. But look at our identities over here. Which one 00:05:52.196 --> 00:06:02.192 does this look like? So we see cosine cosine plus sine sine. Well that looks like this one, doesn't it? So in 00:06:02.194 --> 00:06:14.727 this case, here's my alpha, and here's my beta. So this could be rewritten as the cosine of alpha minus beta. 00:06:14.729 --> 00:06:24.226 So we're going to have 7 pi over 12 minus 5 pi over 12. Well that's the same thing as the cosine, of course, 00:06:24.228 --> 00:06:34.745 of 2 pi over 12, which reduces to the cosine of pi over 6, and of course that's much easier to find. 00:06:34.747 --> 00:06:41.752 Remember the cosine of pi over 6 is simply the square root of 3 over 2. So I've taken this complicated 00:06:41.754 --> 00:06:48.450 expression, recognized it as one of these identities, and rewritten it. 00:06:50.113 --> 00:06:56.611 In this example, we're given that the sine of alpha is 3/5. Alpha is between pi over 2 and pi. The tangent 00:06:56.613 --> 00:07:01.858 of beta is negative square root of 5. Beta is between negative pi over 2 and 0. We want to find the exact 00:07:01.860 --> 00:07:08.858 value of the sine of alpha minus beta and the tangent of alpha plus beta. Now what's important here is 00:07:08.860 --> 00:07:14.360 that we draw the angles in the correct quadrant, so notice that alpha is between pi over 2 and pi, so that 00:07:14.362 --> 00:07:22.798 means that alpha is in quadrant two. So we're going to draw a picture for alpha in quadrant two, 00:07:22.780 --> 00:07:30.297 and we know the sine is 3/5. We know that the sine is y over r, so we know this side is 3. This side is 5. 00:07:30.299 --> 00:07:37.042 We use the Pythagorean theorem. The third side is 4, but we know that since we're in the second quadrant 00:07:37.044 --> 00:07:47.796 the x-coordinate is negative, so the x-value is negative 4. The y is 3, and r is 5. Now we'll do the same 00:07:47.798 --> 00:07:57.330 thing for beta. Beta is between negative pi over 2 and 0. Well negative pi over 2 is here, 0 is here, so we 00:07:57.332 --> 00:08:04.825 know beta is in quadrant four. We're going to draw a picture for beta. The tangent is negative square 00:08:04.827 --> 00:08:09.093 root of 5, so we know the y over x is negative square root of 5. So we know we're going to put square 00:08:09.095 --> 00:08:16.349 root of 5 on the y, and we're going to put 1 on the x, and again we know that in the fourth quadrant the 00:08:16.351 --> 00:08:23.349 x is positive, the y is negative, and of course the hypotenuse is always positive, r is always positive. 00:08:23.351 --> 00:08:31.096 Using the Pythagorean theorem, we get this side to be the square root of 6. And now we've got the 00:08:31.098 --> 00:08:36.095 pictures drawn. We've got all of the sides, so now we're ready to begin writing the identitites. So let's do 00:08:36.097 --> 00:08:44.098 part a. We want to find the sine of alpha minus beta. We have to remember that identity goes sine alpha 00:08:44.100 --> 00:08:53.133 cosine beta minus cosine alpha sine beta, and once we have these values, we can find these values 00:08:53.135 --> 00:08:58.500 pretty easily. The sine of alpha, we make sure that we go to alpha, so it's important that you label them. 00:08:58.502 --> 00:09:10.410 The sine of alpha is 3/5. Cosine of beta we go here, x over r, so it's 1 over the square root of 6. 00:09:10.412 --> 00:09:24.905 Cosine of alpha would be negative 4/5. Remember x is negative here. And the sine of beta is y over r, 00:09:24.907 --> 00:09:31.899 so the sine of beta would be negative square root of 5 over the square root of 6. Now that we've got our 00:09:31.901 --> 00:09:42.653 values we can simply combine. What do we get? We get 3 over 5 square roots of 6 minus 4 square roots 00:09:42.655 --> 00:09:53.194 of 5 over 5 square roots of 6. Since we have a common denominator, we can combine the numerators. 00:09:53.196 --> 00:10:02.000 So our final answer is 3 minus 4 square roots of 5 over 5 square roots of 6. Now let's look at part b. 00:10:03.500 --> 00:10:12.466 Now in part b, we want to find the tangent of alpha plus beta, so again we write the identity for that, 00:10:12.468 --> 00:10:23.214 which is tangent of alpha plus tangent of beta over 1 minus tangent alpha tangent beta. And now that 00:10:23.216 --> 00:10:27.715 we've got our pictures still, we'll just use those to find these values. What's the tangent of alpha? 00:10:27.717 --> 00:10:39.218 Remember tangent of alpha is y over x, so we're going to get 3 over negative 4, and the tangent of beta 00:10:39.220 --> 00:10:53.964 would be y over x which is negative square root of 5. Now we can't leave our answer like that because 00:10:53.966 --> 00:11:00.012 we can't leave complex fractions, so we have to simplify that. Really the easiest way to do that is 00:11:00.014 --> 00:11:08.467 to multiply, in this case numerator and denominator by 4. So let's see what we get. We're going to get 00:11:08.469 --> 00:11:17.959 negative 3, and then we're going to get minus 4 square roots of 5. In the denominator, we're going to 00:11:17.961 --> 00:11:29.540 get 4 and two negatives, so we have minus 3 square roots of 5. We can leave our answer like that. We 00:11:29.542 --> 00:11:37.539 can leave radicals in the denominator. We just can't leave complex fractions, so here's our final answer. 00:11:37.541 --> 00:11:41.290 Now let's look at one final example. 00:11:43.700 --> 00:11:49.807 In this example, we want to find the exact value of the cosine of the inverse tangent of 4/3 plus the 00:11:49.809 --> 00:12:00.068 inverse cosine of negative 9 over 41. We can let the inverse tangent of 4/3, we can call that alpha, and 00:12:00.070 --> 00:12:07.072 we'll call the inverse cosine of negative 9 over 41 beta. Here's where it's very important that you know 00:12:07.074 --> 00:12:14.087 the range of these inverse functions. Remember the inverse tangent lies between negative pi over 2 and 00:12:14.089 --> 00:12:22.341 pi over 2, and the inverse cosine lies between 0 and pi. So you can see it is very important that you know 00:12:22.342 --> 00:12:30.353 that and not have to look it up. So now what we have is the cosine of alpha plus beta. Remember that 00:12:30.355 --> 00:12:40.364 identity. We need to again draw pictures for both alpha and beta. Now alpha is the inverse tangent of 4/3. 00:12:40.366 --> 00:12:46.612 Since this number is positive and alpha lies between negative pi over 2 and pi over 2, we know alpha 00:12:46.614 --> 00:12:56.624 is in quadrant one. So we'll draw a picture for alpha. The tangent is 4/3, so the y is 4, the x is 3, and 00:12:56.626 --> 00:13:04.379 again we can use the Pythagorean theorem to find the third side which will be 5, and we'll do the same 00:13:04.381 --> 00:13:14.627 thing for beta. Beta lies between 0 and pi. This number is negative, so we know it lies in quadrant two. 00:13:14.629 --> 00:13:23.132 Cosine is x over r, so this side is 9. This side is 41, and again we use the Pythagorean theorem to find 00:13:23.134 --> 00:13:32.377 the third side. It turns out to be 40. So now that we have our pictures, we'll write the identity for the 00:13:32.379 --> 00:13:43.419 cosine of alpha plus beta, which is cosine alpha cosine beta minus sine alpha sine beta, and then we 00:13:43.421 --> 00:13:57.900 can just plug in our values. What's the cosine of alpha? It will be 3/5. The cosine of beta will be, let's see. 00:13:57.902 --> 00:14:09.450 Cosine of beta will be negative 9 over 41. Sine of alpha would be 4/5, and sine of beta would be 40 over 41. 00:14:09.452 --> 00:14:14.952 So now that we've plugged in our values, we simply determine these values. So that will give you 00:14:14.954 --> 00:14:32.971 negative 27 over 205 minus 160 over 205, which gives you negative 187 over 205 for our final answer. 00:14:32.973 --> 00:14:38.975 Now in these examples, they illustrate how important it is to know the exact values of the trig functions 00:14:38.977 --> 00:14:44.978 of the special angles as well as the range of the inverse functions. So be sure you know that. 00:14:44.980 --> 00:14:48.475 Then practice these examples until you've mastered them.